The present thesis aims to develop an analogy between the methods for recognizing a black box group and the classification of the finite simple groups. We propose a uniform approach for recognizing simple groups of Lie type which can be viewed as the computational version of the classification of the finite simple groups. Similar to the inductive argument on centralizers of involutions which plays a crucial role in the classification project, our approach is based on a recursive construction of the centralizers of involutions in black box groups. We present an algorithm which constructs a long root SL_2(q)-subgroup in a finite simple group of Lie type of odd characteristic $p$ extended possibly by a p-group. Following this construction, we take the Aschbacher&#039 / s ``Classical Involution Theorem&#039 / &#039 / as a model in the final recognition algorithm and we propose an algorithm which constructs all root SL_2(q)-subgroups corresponding to the nodes in the extended Dynkin diagram, that is, our approach is the construction of the the extended Curtis - Phan - Tits presentation of the finite simple groups of Lie type of odd characteristic which further yields the construction of all subsystem subgroups which can be read from the extended Dynkin diagram. In this thesis, we present this algorithm for the groups PSL_n(q) and PSU_n(q). We also present an algorithm which determines whether the p-core (or ``unipotent radical&#039 / &#039 / ) O_p(G) of a black box group G is trivial or not where G/O_p(G) is a finite simple classical group of Lie type of odd characteristic p answering a well-known question of Babai and Shalev. The algorithms presented in this thesis have been implemented extensively in the computer algebra system GAP.

In this thesis, we will see a way to use representation theory and the theory of linear algebraic groups to characterize certain family of finite groups. In Chapter 1, we see the history of preceding work. In particular, J. G. Thompson’s classification of minimal finite simple nonsolvable groups and characterization of solvable groups will be given. In Chapter 2, we will describe some background knowledge underlying this project and notation that will be widely used in this thesis. In Chapter 3, the main theorem originally conjectured by Guralnick and Tiep will be stated together with the base theorem which is a reduced version of main theorem to the case where we have a quasisimple group. Main theorem explains a way to characterize the finite groups with a composition factor of order divisible by two distinct primes p and q as the finite groups containing nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In this thesis we more focus on the proof of showing a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In Chapter 4, we will prove a set of lemmas and proposition which will be used as key tools in the proof of the base theorem. In Chapters 5 to 7, we will establish the base theorem in the cases where a quasisimple group G has its simple quotient isomorphic to alternating groups or sporadic groups (Chapter 5), classical groups (Chapter 6), and exceptional groups (Chapter 7). In Chapter 8, we show that any finite group G admitting nontrivial 2-element x, p- element y, q-element z such that xyz = 1 for two distinct odd primes p and q admits a composition factor of order divisible by pq. Also, we show that the question if a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1 can be reduced to the base theorem.

In this thesis I study metric aspects of finite nearly simple groups. Its four distinct chapters deal with four different questions. In the first chapter, I give a full description of the normal subgroup lattice of any algebraic ultraproduct of universal finite quasisimple groups. In the second, I investigate approximation questions for arbitrary abstract and topological groups by families of finite groups with conjugacy-invariant norms. In the third chapter, I prove that the map induced by any non-trivial word on the metric ultraproduct of classical groups of Lie type of unbounded rank is always surjective using cohomological and algebraic methods. In the last chapter, it is proved that (simple) metric ultraproducts of finite classical groups of Lie type of unbounded rank with different field sizes are always non-isomorphic. Also, if the field sizes are equal, two such ultraproducts can only be isomorphic if the Lie types are equal or one Lie type is orthogonal and the other symplectic.:Introduction 0 Notation, basic definitions, and facts 0.1 Group theory 0.2 Some ring and field theory 0.3 Ultraproducts and norms 1 The normal subgroup lattice of an algebraic ultraproduct 1.1 Introduction 1.2 Auxiliary geometric results 1.3 Relative bounded normal generation in universal finite quasisimple groups 1.4 The lattice of normal subgroups 2 Metric approximation of groups by finite groups 2.1 Introduction 2.2 Preliminaries 2.2.1 On C-approximable abstract groups 2.2.2 On C-approximable topological groups 2.3 On Sol-approximable groups 2.4 On Fin-approximable groups 2.5 On the approximability of Lie groups 3 Word maps are surjective on metric ultraproducts 3.1 Introduction 3.2 Symmetric groups 3.2.1 Power words 3.2.2 The cycle structure of elements from PSL_2(q) 3.2.3 Effective surjectivity of word maps over finite fields 3.2.4 Proof of Theorem 3.1 3.3 Unitary groups 3.3.1 Proof of Theorem 3.3 3.3.2 Further implications 3.3.3 Concluding remarks 3.4 Finite groups of Lie type 3.4.1 The linear case 3.4.2 The case of quasisimple groups of Lie type stabilizing a form 3.4.3 An alternative way of proving Theorem 3.1 using wreath products 4 Isomorphism questions for metric ultraproducts 4.1 Introduction 4.2 Description of conjugacy classes in S_U, GL_U(q), and PGL_U(q) 4.3 Characterization of torsion elements in S_U , GL_U(q), and PGL_U(q) 4.4 Faithful action of S_U and PGL_U(q) 4.5 Centralizers in S_U , GL_U(q), Sp_U(q), GO_U(q), and GU_U(q) 4.6 Centralizers in PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.7 Double centralizers of torsion elements 4.7.1 The case S_U 4.7.2 The case PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.8 Distinction of metric ultraproducts 4.8.1 Computation of e_G(o) when gcd{o,p}=gcd{o,|Z|}=1 4.8.2 Proof of Theorem 4.1 Index of Symbols Index Bibliography / In dieser Doktorarbeit studiere ich metrische Aspekte von endlichen fast-einfachen Gruppen. Ihre vier Kapitel beschäftigen sich mit vier unterschiedlichen Themenfeldern. Im ersten Kapitel gebe ich eine vollständige Beschreibung des Normalteilerverbandes eines algebraischen Ultraproduktes von universellen endlichen quasieinfachen Gruppen. Im zweiten beschäftige ich mich mit Approximationsfragen für beliebige abstrakte und topologische Gruppen durch Familien von endlichen Gruppen, auf denen eine konjugationsinvariante Norm erklärt ist. Im dritten Kapitel beweise ich, dass die Abbildung auf einem metrischen Ultraprodukt von klassischen Gruppen vom Lie-Typ von unbeschränktem Rang, die von einem beliebigen nicht-trivialen Wort induziert wird, immer surjektiv ist. Dabei verwende ich sowohl kohomologische als auch algebraische Methoden. Im letzten Kapitel beweise ich, dass (einfache) metrische Ultraprodukte von klassischen endlichen Gruppen vom Lie-Typ von unbeschränktem Rang mit unterschiedlicher Körpergröße immer nicht-isomorph sind. Ist die Körpergröße gleich, so können zwei solche Gruppen nur dann isomorph sein, falls sie auch denselben Lie-Typ haben, oder eine vom orthogonalen Typ und die andere vom symplektischen ist.:Introduction 0 Notation, basic definitions, and facts 0.1 Group theory 0.2 Some ring and field theory 0.3 Ultraproducts and norms 1 The normal subgroup lattice of an algebraic ultraproduct 1.1 Introduction 1.2 Auxiliary geometric results 1.3 Relative bounded normal generation in universal finite quasisimple groups 1.4 The lattice of normal subgroups 2 Metric approximation of groups by finite groups 2.1 Introduction 2.2 Preliminaries 2.2.1 On C-approximable abstract groups 2.2.2 On C-approximable topological groups 2.3 On Sol-approximable groups 2.4 On Fin-approximable groups 2.5 On the approximability of Lie groups 3 Word maps are surjective on metric ultraproducts 3.1 Introduction 3.2 Symmetric groups 3.2.1 Power words 3.2.2 The cycle structure of elements from PSL_2(q) 3.2.3 Effective surjectivity of word maps over finite fields 3.2.4 Proof of Theorem 3.1 3.3 Unitary groups 3.3.1 Proof of Theorem 3.3 3.3.2 Further implications 3.3.3 Concluding remarks 3.4 Finite groups of Lie type 3.4.1 The linear case 3.4.2 The case of quasisimple groups of Lie type stabilizing a form 3.4.3 An alternative way of proving Theorem 3.1 using wreath products 4 Isomorphism questions for metric ultraproducts 4.1 Introduction 4.2 Description of conjugacy classes in S_U, GL_U(q), and PGL_U(q) 4.3 Characterization of torsion elements in S_U , GL_U(q), and PGL_U(q) 4.4 Faithful action of S_U and PGL_U(q) 4.5 Centralizers in S_U , GL_U(q), Sp_U(q), GO_U(q), and GU_U(q) 4.6 Centralizers in PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.7 Double centralizers of torsion elements 4.7.1 The case S_U 4.7.2 The case PGL_U(q), PSp_U(q), PGO_U(q), and PGU_U(q) 4.8 Distinction of metric ultraproducts 4.8.1 Computation of e_G(o) when gcd{o,p}=gcd{o,|Z|}=1 4.8.2 Proof of Theorem 4.1 Index of Symbols Index Bibliography